Slide rule



Aug. l, 1939. c. M. BERNEGAU SLIDE RULE Filed Aug. 20, 1937 Patented Aug. 1V, 1.939 l i UNITED STATESPATENT OFFICE SLIDE RULE Carl M. Bernegau, Hoboken, N. J., assigner to Keuffel & Esser Company, Hoboken, N. J., a corporation of New Jersey Application August 20, 1937, Serial No. 160,016

6 Claims. (Cl. 235-70) This invention relates to slide rules of the kind quired a different method of manipulating a rule in which ndices in the form of scales adapted so that great difliculty was experienced by all to coact with one another in accordance with users in determining the proper manipulation well known laws are arranged on the respective to be used in solving problems with which they relatively movable component elements. had not had recent and extensive practice. No 5 Three types of slide rules are in general use. slide rule heretofore proposed has ever overcome In one type, relatively movable elements reciprothis diiiiculty, cate with respect to one another and on the re- In copending application Serial No. 137,400 spective elements are parallel coacting scales. led April y17, 1937, there is provided a slide rule, lo Such slide rules comprise a body member coinirrespective of type, by which, with the applical@ posed of spaced parallel side bars between which y tion of not more than two easily understood and reciprocates, as by means of a tongue and groove readily memorized principles, the user is able to connection, a slide, the scales appearing on one devise the best settings for any particular purpose or both surfaces of the rule. Another type of and to recall settings which have been forgotten l5 slide rule involves concentric rotatable discs 0I" In this slide rule problems involving numerical i5 progressively varying diameter around the peand trigonometric terms may be solved, irrespec- TillhelieS Of Which are llfalgd the COCtiilg tive of the number of steps therein, in a continscales. So-called cylindrical slide rules consist, nous manipuiation, that is, Without the necessity usually, in an axially rotatable cylinder on the of having to Set down a result in Order ts retain surface of which are scales extending in the lonthe Same While a new Setting is mada 20 gitudinal direction of the rule while a rotary and An Object of the present invention is to facili.. usually axially movable slide concentric theretate the reading of the trigonometric Scales. with carries parallel coacting scales. In all these Another Object of the invention is to facilitate l'ilieS iii@ indices 0f Scales are arranged m Il )g the reading of the trigonometric scales in the arithmic proportions, the arrangement being aforesaid Continuous manipulation. 5

based upon the principle that the sum of the logarithms of numbers is equal to the logarithm of the product of those numbers. To make computations in multiplication, for instance, it has been the practice to bring the index of one log arithmic scale to the scale divisions representing the logarithm of one of the numbers to be mul- In carrying the invention into effect, one of two relatively movable members of the slide rule has a scale which is graduated in scale graduations to give readings of angles of the trigonometric function which are found by reference to a logarithmic scale of the same unit length on tipied on a Coaching scale and then read the the other member, the indicia of the rst'narned product at the scale division representing the log ficale .graduatlons bffmg slanted m the dlrecilOn of that product opposite the scale division on the m which the ?ng 1e 1nc1`ase5 In respect 0f the 35 moved scale representing that number by which tafngeit S0219, mdlCia are DIOVided Which Slant in it is multiplied. Division is performed as a rethe dlectiqn in Whlh the angleincreases While Verse 0f this manipuiation other indicla slant 1n the opposite direction to Simple computations have been thus readily indicate the continuation of such scale. Also, if effected by movement of the slid@ with respect desired, the color of one series of the indicia may to the body and, as the art has progressed, addicontrast with the color of another series of intional coacting scales have been added to slide dicia. rules enabling the solution of problems in square These and other objects of the invention and and cube root, in trigonometry and in the fgurthe means for their attainment will be more aping of fractional powers and roots. A slide rule parent from the following detailed description, 45

has been devised by which the right triangle has baken in COIiIleCtOrl With the aCCOmDaIlYiIIE been solved by a single setting of the rule but in diaWil'lg illustrating 0D@ embOdiment by which the solution of other problems a sequence of bhe iIiVeIltOIi may be realized and in which:

steps has been necessary, each step necessitating Figure 1 iS 21 P1211 VieW Showing one face of the to the proper relation of slide and body and the Slide ruiathe Central portion beine broken away; 5o making oi a notation of the results found after Figure 2 is a similar view showing the reverse each step in order that the parts may be again face; manipulated in the performance of a further Figure 3 is a fragmentary view on an enlarged step using the result found in the previous comscale showing the face of the slide of Figure 2; putations. Each individual problem further re- Figure 4 shows a triangle lllustrative of the 55 solution of various problems by the rule of this invention; and

Figure 5 shows a closed figure, problems in respect of which may be similarly solved.

In the drawing, side bars H and J are rigidly secured together by means of plates P which are Aus secured thereto at M, so that a sliding bar N may be mounted between said bars H and J so as to be readily slidable longitudinally thereof. The slidable bar N has the usual tongue members on eachged'geadapted to slide in the usual groove members on theedges of the bars H and J contiguoustofthe sliding bar N so that the sliding bar N is always-heldin engagement between the bars- Hand. J5 in. whatever position it occupies longitudinally thereof. A runner or indicator X, transparent onbothffaces and of any usual constructiomisrnounted over: said bars H, J and N so aste-'be 'readily movedinto any position desiredbetween the plates P,"andthe runner X haga hairlinev Y on each side thereof.

.On the front side otjthe rule,- as shown in Figure 1, the 'pper scale on .the bar-.H is vdesignatedg'as L andisa scale of equal parts from Q to 1.0 and'isused to obtain common logarithms when treten-red, as will vbe understood, tothe D scale on bar J, referred tOEhereinaiter.. The scale next be. "lowfthe Il' scale on this faceis designated as LLI` and :ha'sfgraduations representing logarithms .of

thelogarithmsfoi'. thenumbers 11.01 to 1.11. .The

'scalebelow the `LLI scale is designed .as DF, and is a standard -lo'garithmicscaleof full Dit length theisarne asi'thefDfscale whichv is yet tolbe described,fexceptthat it isfolded and has its vindex atthe center. This scale is proximate the inner The frontgface, as viewed, or the slide is .pro-

vided along its `upper edge A,with a scale desgnatedas CF and is identical'withthe. vDF scale, on the upperl side bar Alsocn the front face' of the slide below the' CFscale isa 'scale designated as CIF, .which is a standard: folded reciprocal logarithmic. scale of f ull unit length graduatedl from 10.-to 1 similar in' every respecttoy th'eD scale soon tobe described except vthat .it is .foldedand inverted. Immedi-v at'elyA below-the' CIF. scale is thefscale` designated as CI whichl is a'v standard. reciprocal. logarithmic scale ofjfullunit length graduated from.10 tol. Below the CI scaleand'proximate-the lower edge of the slide is' a fourthscale designatedas C and has standard graduated logarithmicdivisions f of full unit length from 1 tc.. 10.xy This scaleis the same as the D scale on the body. (bar J) next to' f be described. V

On the upperedge ofthe-side bar .I proximate the slide is a scale designated as D, which is the,- same as the C scale and is a standardvlogarithmic scale of full unit length. Along thelower edgev of the side bar J is a scale designated LL! which has graduations representing the logarithms of-4 logarithms of the numbers from 1.11 to 2.7. The middle scale on the front face of the lower side bar is a scale, designated as LL3, which has grad" uations representing the logarithms of logarithms of the numbers from 2.7 to 22,000. These three scales LLI, LL2 and LL3 are, conveniently, continuations of `ea'ch other, as will be clear from the foregoing.

On the rear face of the slide rule, as shown in Figure 2, is a scale designated LLM) along the upper edge of the upper (as viewed) side bar J and immediately below that scale LL00 is a scale designated as LLB representing the logarithms of co-logarithms between 0.00005 and 0.999. These scales are used to nd powers and roots 'of numbers below unity and to iind cologarithms to any base of numbers below unity. The lower scale on the upper side bar J, on the rear face, is

designated A and comprises a standard graduated logarithmic scale of two unit lengths from 1 to 10. The A scale represents the natural co-logarithms of the numbers on the LLB and LL00 scales.

On the slide N, on this rear face of the rule, and along the upper edge thereof, is a scale designated as B which has the same graduations as the scale A. Immediately below the B scale on the slide is a scale designated as T whichis a tangent scale with divisions to represent angles from 545' to 45. This range of angles will give tangents in connection with the C or D scale and cotangents in connection with the CI or DI scale. As shown in Figure 3, the numerals G representing the angles below 45 are indicated inl one color, say, black, and slanted in the direction in which the scale is read, While the numerals Rindicating the continuation of this scale (45 to 8415) in the opposite direction are in a different color, say, red, and slant in the opposite direction, i. e., the direction in which the angle increases and in which this scale is read'.` ,The angles noted in red will give tangents in connection with the CI or DI scales and cotangents in connection with the C or D scales.

Immediately below the tangent scale T is a scale designated as ST. This scale is used whenever'an angleless than 5 44', that is, an angle Whosesine oretangevnt is less than 0.1, is involved in the solution-oi?v aVtrigonometri'cal problem. 'I'he y lowermost scale orrfthe slide N, on the rear face for the cosine to beobtainedjon the CI or-.DI scales when reading, however, in the opposite. direction.

therefore) in onecolor, say, black; and slant in the .The numerals U used in obtaining the sine. are,`

. li5-- vdirection -inwhich the -angle increases, .whilev thenllneralsfW-used in vobtaining the cosine are in a dierent-and contrastingcolor, say, redand slantin anoppositeldirection, i. el, in the direction Ain which. the angle-,increases vand in which this scale islread.v

the bar H), andy along Vthe'ed'ge proximatev the slide N;is another scalegdesignatedjas D whichis` exactly the ,same as theDfscale onv the front face hereinbeforedes'cribd. 'Below `'theD scaleis a.. scaleldesignatedas DI .andis ajstandard'reciprocallogarithmic scale of full unitlength graduated fron'rlO to. `1 the sameaslthe D-'scale except that '60' it is inverted.; The lowermost scale onsthisv face v is-avsc'ale designated as K. i This scale isgradu` atedito-show the cube of a number inthe same transverse line.v on .the D 'scale and ccmprisesthe standard graduated.logarithmic'scale of .threel unit lengths.. Conversely, ofcourse, the Dscaleshows the cube root ci the corresponding graduation on the K scale'. It will be notedthat there' are three partstothe K scale, veach the samenas the others exceptl as to position'. These-parts will be referred to hereinafteras vthe left hand part,

the middle partv and the right-hand part, or"k as K left, K middle and K right,v respectively. The cube root of a given number is a second'number'ivhose .v

cube is the given number.

In order to understand the simplicity vot oper'- ation of this slide rule, as explained hereinbefore, whereby the operation thereof, irrespective of the number of steps, may be carried out continuously without resetting the parts to a result found in the previous step, the principle of operation for such simple problems as were possible with prior slide rules will first be explained, leading into more complicated problems which are only possible with the slide rule of this invention, in order to explain that the same simple procedure may be applied to all problems whereby the user, having mastered the fundamental rules, is now able with this rule to perform all problems as they arise.

In the following illustration for the sake of brevity, the various scales are referred to merely by their letters such as C, D, etc., instead of C scale, D scale and the like:

Example 1 Evaluate Solution: The user reasons as follows: first divide 73.6 by .5 and then multiply the result by 3.44, This would suggest Push hairline to 73.6 on D, Draw .5 on C under the hairline, Push hairline to 3.44 on C and Read 506 on D scale.

Example 2 Evaluate 18 45 37 23)(29 Solution: Reason as follows: (a) divide 18 by 23, (b) multiply the result by 45. (c) divide this second result by 29, (d) multiply this third result by 37. This suggests to Push hairline to 18 on D Draw 23 on C to hairline,

Push hairline to on C,

Draw 29 on C to hairline,

Push hairline to 37 on C and Read 449 on D scale. To determine the decimal point in the answer, a rough approximation is made. For the above example write Hence the answer is 44.9.

There are other slide rules by means of which the foregoing examples can be solved as readily as on the slide rule of this invention. Their presentation here is only for the purpose of showing that the same principle and the same procedure used in solving these relatively simple problems are also applicable for solving problems involving trigonometric functions on this slide rule. This is possible due to the fact that all trigonometric scales are of a common full CD unit length and that they are rendered more flexible by being placed on the slide.

The following group of examples exemplify this:

Example 3 Evaluate 73.6 3.44 sin 30o Solution:

To 73.6 on D scale Draw 30 on S scale,

Push hairline to 3.44 on C scale and Read 506 on D scale.

By comparing this solution with that of Example 1, it will be seen that the two are identical.

Example 4 Evaluate 53.5 2.25 sin 1330 Solution:

2.25 sin 133o=sin 133o To 53.5 on D scale Draw 1330' on S scale, Push hairline to 2.25 on CIF scale and Read 101.9 on DF scale.

Example 5 Evaluate 17 cos 5030l tan 12 sin 59 Solution (compare with Example 2):

17 cos 5030' 17 coa 5030Xl tan 12 sin 59'- ran 12 sin 59' To 17 on D scale Draw 12 on T scale,

Push hairline to 30 on S (red), Draw 59' on ST scale to hairline,

Push hairline to 1 (index) of C scale and Read 2960 on D scale.

Example 6 Evaluate 1/2-.9 cos 16 tan 16 Solution:

To 2.9 on A scale Draw 16 on T scale,

Push hairline to 16 on S (red) and Read 5.71 on D scale.

plained in the copending application, it would l be necessary to transpose one factor from the denominator to the numerator in form of its reciprocal.' But there are no inverted trigonometrical scales on the rule. However, by inserting unity twice in the numerator, we will have the desired condition. Thus:

'can 15 1X tan 15 1 cos 20 sin 25-cos 20 sin 25 To 1 on D scale Draw 20 on S (red), Push hairline to 15 on T scale, Draw 25 on S scale to hairline,

` Push hairline to 1 on C scale and Read .675 on D scale.

Example 8 Evaluate 22 cos2 33 93 Solution:

To 22 on A scale Draw 93 on B scale,

Push hairline to 33 on S (red) and Read .166 on A scale.

Example 9 Example 11 Evaluate Evaluate 8.2 2 tem2 21 -gc1j;g2 45 Solutlon' Solution:

T0 8-2 0n C Scale To 1.4 on LLZ scale Draw 40 0n S Scale- Draw '10 on s (red) Push hairline to 21 on T scale and Push indicator to 45on s scale and Read 2395 0n A scale- Read .6955 on D scale.

s Example 10 Example 12 Evaluate Evaluate D l O l 3s65\/7.8s 7s sin 38238654733 sin 3s 1 cos 79 05 loinlgs 8C 7 1 tan 36 cos 25 (1/75) tan 36 cos 25 Solution:

To 3865 on D scale Draw on CI scale,

Push hairline to 7.83 on B left, Draw 36 on T under hairline Push hairline to 38 on S,

Draw 25 on S (red) under hairline, At index of C scale,

Read 758,800 on D scale.

These examples show that the same settings for simple multiplication and division on the C andD scales are also used when trigonometric functions are introduced. To evaluate the expression of Example 10 on a rule with trigonometric scales of unequal unit lengths or a rule where trigonometric scales are placed on the body of the rule, one would probably rst replace two of the trigonometric functions by their actual numerical values and then proceed as above. Much of the simplicity of operation of this slide rule consists in the fact that a single type of setting applies'without exception in the case of combined multiplication and division problems whether they are simple or complex.

The arrangement of scales of this rule permit the use of the same law for solving triangles wherever possible without reverting to the opposite side of the rule. Thus to solve the triangle shown in Figure 4:

To 381 on D scale Draw 66 on S scale, -v

Push hairline to 42 on S scale And read a==279 (answer) on D scale; Push hairline to 72 on S scale An'd read b=396 (answer) on D scale.

Solution:

To 362 on D scale Draw 52 on S scale,

Push hairline to on S, Draw 40 under hairline, Push hairline to 72 on S, Draw 90 on S under hairline, Push hairline to 70 on S And read h=637 on D! The process consists of a succession of movements of the slide and of the hairline with a nal reading of the desired quantity. To solve the same problem on a rule with trigonometric scales located on the body, the three triangles would have to be solved separately by reading the sides marked a: and y as intermittent results.

Solution:

To 1.0498 on LL1 Draw 55 on S,

Push hairline to 7905 on S (reg) Draw 7010' on S (red) to hairline and Read at the index .03319 on D.

Example 13 Evaluate 416.2 10g. 135 csc 60 Solution:

To on LLB scale Draw 60 on S scale, Push hairline to 16.2 on B and Read 22.8 on D scale.

the two solutions will be clearly seen.

Example 3 73.6X3.44 sin 30 Solution:

To 73.6 on scale A Set 30 on scale S At 3.44 on scale B Read 506 on scale A.

'I'his solution is the same as on the slide rule of this invention, with the exception that, on such slide rule, scales of 25 c/m unit length can be used while on the rule ofthe patent, scales of only 12.5 c/m unit length have to be used. This naturally tends to make the answer more accurate on the rule of this invention.

' Example 4 Evaluate v 2.25 sin 13030 Solution:

To 53.5 on Ascale Set 1330' on S scale Hairline to left index of B scale Set 2.25 on B scale under hairline And opposite index on B scale Read 101.9 on A scale.

Two movements of the slide are required on 12.5

c/m unit scales, while on the slide rule of this 75 invention only one movement of the slide is necessary. The reason is that there 25 c/m unit scales are used, and, therefore, advantage of the inverted scales could be taken.

Example 5 Evaluate 17 cos 5031 tan 12 sin 59 Solution: The trigonometric scales of the rule of the patent are not numbered for complements; therefore cos 5030=sin (90-5030)=sin 3930 Since the T scale is of different unit length, it can not be used in conjunction with the S scale. Therefore, the natural tangent of 12 must first be found:

Opposite 12 on T scale Read .2125 on C scale.

The problem now is 17 sin 39030 .2125 sin 59 To 17 on A scale Set .2125 on B scale Hairline to 3930' on S scale; 59 on S scale to hairline At index of B scale Read 2960 on A scale.

Usually a user will prefer to find the numerical values of all the trigonometric functions in the computation and then proceed as in regular multiplication and division, in order to use 25 c/m unit scales.

Example 6 V2.9 cos 16 The example now is J2.9 .96 tan 16 To 2.9 on Ascale Set 16 on T scale Opposite .96 on C scale Read 5.7 on D scale.

Evaluate Example 7 Evaluate tan 15 cos 20 sin 25 Solution:

At 15 on T scale Read .268 on C scale.

At (90-20) :70 on S scale Read .94 on B scale.

At 25 on S scale l Read .423 on B scale.

The example now is To .268 on D scale Set .423 on C scale At .94 on CIF scale Read .674 on DF scale.

It is easily seen that the solution on the rule o! this invention is much simpler and quicker.

Example 8 Evaluate 22 cos 33 93 Solution:

Cos2 33=sin2 (90-33)=sin 57 At 57 on S scale Read .839 on B scale.

The problem now is To 22 on A scale Set 93 on B scale. At .839 on C scale Read .166 on A scalef` Example 9 Evaluate (8.2)2 tan2 21 sin2 40 Solution:

At 40 on S scale Read sin 40=.642 on B scale. To 8.2 on D scale Set .642 on C scale.

At 21 on T scale Read 24 on A scale.

Example 10 Evaluate `38651/7153 75 sin 38 tan 36 cos 25 Solution: Again rst nd the numerical values of the trigonometric functions, and then proceed With multiplication and division.

At 38 on S scale Read .616 on B scale.

At 90-25=65 on S scale Read .906 on B scale.

At 36 on T scale Read .726 on D scale.

The problem now is:

ssen/7.83 75 .616 .725x .906

This is solved in the customary way.

It is seen that in almost every problem in which trigonometric functions occur either in the numerator or in the denominator, the solution on the rule of this invention will be much simpler and the result in most cases more accurate than on the rule of the patent due to the fact that all trigonometric scales are of the same unit length and may be, for instance, 25 c/m long, which permits them to work in conjunction With all the other scales on the rule.

Example 11 Evaluate loge 1.4 sin 45 cos 70 Solution: The natural logarithms of numbers above unity are found on LL3, LL2, and LLI. These scales are of 25 c/m unit lengths while the S scale is of 12.5 c/m unit length. Therefore the natural functions of sin y45 and cos '70 have to be inserted in the above problem. y

Cos 70=sin (90-70) =sin 20 Opposite 20 on S scale Read .342 on B scale. Opposite 45 on S scale Read .707 on B scale.

The problem now reads:

To 1.4 on LLI scale Set .342 on C scale. Opposite .707 on C scale Read .695 on D scale.

Eample 12 Evaluate cos 7905 log. 1.0498 sec 7010' sin 55 Solution: Here again all the trigonometric functions are found on a 12.5 c/m unit length scale, namely the S scale. Therefore their numerical value has to be found first.

The problem now is:

.1895 loge 1.0498 .82X.34

To 1.0498 on LL1 scale Set .82 on C scale.

Hairline to .1895 on C scale,

.34 on C scale to hairline At index of C read .033 on D scale.

Example 13 Evaluate -\/16.2 log, 135 csc 60 Solution:

o csco sin 60 At 60 on S scale Read .866 on B scale. To 135 on LL3 scale Set .866 on C scale At 16.2 on B scale Read 22.75 on D scale.

It will thus be seen that a slide rule has been provided in which one of the members is graduated in scale graduations to give readings of angles, the trigonometric function of which is found by reference to a logarithmic scale, the indices of the scalel graduations on the first member being slanted in the direction in which the angle increases t'o facilitate the visual selection of the desired scale graduation, the respective colors of the indicia of complementary angles being contrasting.

Various modifications will occur to those skilled in the art in the type of slide rule employing this invention as well as in the selection and/or disposition of the scales and no limitation is intended by the phraseology of the foregoing specification or illustrations in the accompanying drawing eX- cept as indicated by the appended claims.

What is claimed is:

1. A slide rule comprising two relatively movable members, one Aof said members being graduated in scale graduations to give readings of angles the trigonometric function of which is found by reference to a logarithmic scale of the same unit length on the other member, the indicia of the scale graduations on the first member being slanted in the direction in which the angle increases.

2. A slide rule comprising two relatively movable members, one of said members having a scale graduated in scale graduations to give readings of angles the trigonometric function of which is found by reference to a. logarithmic scale on the lother member, said graduations being provided with indicia giving the angle the sine of which is found on said logarithmic scale, said indicia slanting in the direction in which the angle increases.

3. A slide rule comprising two relatively movable members, one of said members having a scale graduated in scale graduations to give readings of angles the trigonometric function of which is found by reference to a logarithmic scale on the other member, said graduations being provided with indicia giving the angle the sine of which is found on said logarithmic scale, said indicia slant'- ing in the direction in which the angle increases and other indicia giving the angle of the cosine thereof and slanting in the direction in which the scale is read for the cosine of the angle.

4. A slide rule comprising two relatively movable members, one of said members having a scale graduated in scale graduations to give readings of angles the trigonometric function of which is found by reference to a logarithmic scale on the other member, said graduations being provided with indicia giving the angle the tangent of which is found on said logarithmic scale, said indicia slanting in the direction in which the angle increases.

5. A slide rule comprising two relatively movable members, one of said members having a scale graduated in scale graduations to give readings of angles the trigonometric function of which is found by reference to a logarithmic scale on the other member, said graduationsy being provided with indicia giving the angle the tangent of which is found on said logarithmic scale, said indicia slanting in the direction in which the angle increases and other indicia slanting in the opposite direction to indicate the continuation of said scale.

6. A slide rule comprising two relatively movable members, one of said members being graduated in scale graduations giving readings of angles and compliments of the angles, the trigonometric function of which is found by reference to a logarithmic scale of the sameunit length on the other member, the indicia of the scale graduations on the rst member being slanted in the direction in which the angle increases.

CARL M. BERNEGAU.

Certificate of Correction 7.

Patent No. 2,168,056. Aug. 1, 1939.

CARL M. BERNEGAU It is hereby certified that error appears in the printed specification of the above numbered patent requiring correction as follows: Page 5, first column, line 13, Example 5, for that portion of the formula reading cos 50 30=sin read cos 50 30- Y sin; and that the said Letters Patent should be read with this correction therein that the same may conform to the record of the case in the Patent Oice.

Signed and sealed this 12th day of September, A. D, 1939.

[SEAL] HENRY VAN ARSDALE, p

Acting Commissioner of Patents.

Certicate of Correction Patent No. 2,168,056. Aug. 1, 1939.

' CARL M. BERNEGAU It is hereby certified that error appears in the printed specification of the above numbered patent requiringcorrection as follows: Page 5, first column, line 13, Example 5, for that portion of the formula reading cos 50 30'-sin read cos 50 30- sin; and that the said Letters Patent should be read with this correction therein that the same may conform to the record of the case in the Patent Oice.

Signed and sealed this 12th day of September, A. D. 1939.

[SEAL] l HENRY VAN ARSDALE, K

Acting Commissioner of Patents. 

